# greatest common divisor of the least values of m and n is

Question:

If $\left(\frac{1+i}{1-i}\right)^{m / 2}=\left(\frac{1+i}{i-1}\right)^{n / 3}=1,(m, n \in \mathbf{N})$, then the

greatest common divisor of the least values of $m$ and $n$ is_________.

Solution:

Given that $\left(\frac{1+i}{1-i}\right)^{m / 2}=\left(\frac{1+i}{i-1}\right)^{n / 3}=1$

$\Rightarrow\left(\frac{(1+i)^{2}}{2}\right)^{m / 2}=\left(\frac{(1+i)^{2}}{-2}\right)^{n / 3}=1$

$\Rightarrow i^{m / 2}=(-i)^{n / 3}=1$

$m$ (least) $=8, n$ (least) $=12$

$\operatorname{GCD}(8,12)=4$